Abstract

A random map is a discrete-time dynamical system where one of a number of transformations is selected randomly and applied in each iteration of the process. In this thesis we study existence, approximation and properties of absolutely continuous invariant measures (acim) for random maps and obtain several new results. We generalize a result of Straube, which provides a necessary and sufficient condition for existence of an acim of a nonsingular map, to random maps. We approximate absolutely continuous invariant measures for Markov switching position dependent random maps using Ulam's method. For certain random maps, we prove the existence of ergodic infinite acims. Finally, we prove that the invariant density of an acim for random maps is strictly positive on its support.